1//===-- Implementation of exphk function ----------------------------------===//
2//
3// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
4// See https://llvm.org/LICENSE.txt for license information.
5// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
6//
7//===----------------------------------------------------------------------===//
8
9#include "exphk.h"
10#include "src/__support/CPP/bit.h"
11#include "src/__support/common.h"
12#include "src/__support/fixed_point/fx_bits.h"
13
14namespace LIBC_NAMESPACE {
15
16namespace {
17
18// Look up tables for exp(hi) and exp(mid).
19// Generated with Sollya:
20// > for i from 0 to 89 do {
21// hi = floor(i/8) - 5;
22// m = i/8 - floor(i/8) - 0.5;
23// e_hi = nearestint(exp(hi) * 2^7) * 2^-7;
24// e_mid = nearestint(exp(m) * 2^7) * 2^-7;
25// print(hi, e_hi, m, e_mid);
26// };
27// Notice that when i = 88 and 89, e_hi will overflow short accum range.
28static constexpr short accum EXP_HI[12] = {
29 0x1.0p-7hk, 0x1.0p-6hk, 0x1.8p-5hk, 0x1.1p-3hk, 0x1.78p-2hk, 0x1.0p0hk,
30 0x1.5cp1hk, 0x1.d9p2hk, 0x1.416p4hk, 0x1.b4dp5hk, 0x1.28d4p7hk, SACCUM_MAX,
31};
32
33static constexpr short accum EXP_MID[8] = {
34 0x1.38p-1hk, 0x1.6p-1hk, 0x1.9p-1hk, 0x1.c4p-1hk,
35 0x1.0p0hk, 0x1.22p0hk, 0x1.48p0hk, 0x1.74p0hk,
36};
37
38} // anonymous namespace
39
40LLVM_LIBC_FUNCTION(short accum, exphk, (short accum x)) {
41 using FXRep = fixed_point::FXRep<short accum>;
42 using StorageType = typename FXRep::StorageType;
43 // Output overflow
44 if (LIBC_UNLIKELY(x >= 0x1.64p2hk))
45 return FXRep::MAX();
46 // Lower bound where exp(x) -> 0:
47 // floor(log(2^-8) * 2^7) * 2^-7
48 if (LIBC_UNLIKELY(x <= -0x1.63p2hk))
49 return FXRep::ZERO();
50
51 // Current range of x:
52 // -0x1.628p2 <= x <= 0x1.638p2
53 // Range reduction:
54 // x = hi + mid + lo,
55 // where:
56 // hi is an integer
57 // mid * 2^3 is an integer
58 // |lo| <= 2^-4.
59 // Then exp(x) = exp(hi + mid + lo) = exp(hi) * exp(mid) * exp(lo)
60 // ~ exp(hi) * exp(mid) * (1 + lo)
61 // with relative errors < |lo|^2 <= 2^-8.
62 // exp(hi) and exp(mid) are extracted from small lookup tables.
63
64 // Round-to-nearest 1/8, tie-to-(+Int):
65 constexpr short accum ONE_SIXTEENTH = 0x1.0p-4hk;
66 // x_rounded = floor(x + 1/16).
67 short accum x_rounded = ((x + ONE_SIXTEENTH) >> (FXRep::FRACTION_LEN - 3))
68 << (FXRep::FRACTION_LEN - 3);
69 short accum lo = x - x_rounded;
70
71 // Range of x_rounded:
72 // x_rounded >= floor((-0x1.628p2 + 0x1.0p-4) * 2^3) * 2^-3
73 // = -0x1.6p2 = -5.5
74 // To get the indices, we shift the values so that it start with 0.
75 // Range of indices: 0 <= indices <= 89
76 StorageType indices = cpp::bit_cast<StorageType>((x_rounded + 0x1.6p2hk) >>
77 (FXRep::FRACTION_LEN - 3));
78 // So we have the following relation:
79 // indices = (hi + mid + 44/8) * 8
80 // That implies:
81 // hi + mid = indices/8 - 5.5
82 // So for lookup tables, we can use the upper 4 bits to get:
83 // exp( floor(indices / 8) - 5 )
84 // and lower 3 bits for:
85 // exp( (indices - floor(indices)) - 0.5 )
86 short accum exp_hi = EXP_HI[indices >> 3];
87 short accum exp_mid = EXP_MID[indices & 0x7];
88 // exp(x) ~ exp(hi) * exp(mid) * (1 + lo);
89 return (exp_hi * (exp_mid * (0x1.0p0hk + lo)));
90}
91
92} // namespace LIBC_NAMESPACE
93

source code of libc/src/stdfix/exphk.cpp